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Keywords:
block designs; orbits; projective linear group; projective special linear group; twisted projective linear group; Kramer-Mesner method
block designs; orbits; projective linear group; projective special linear group; twisted projective linear group; Kramer-Mesner method
Summary:
Using the Kramer-Mesner method, $4$-$(26,6,\lambda)$ designs with $PSL(2,25)$ as a group of automorphisms and with $\lambda$ in the set $\{30,51,60,81,90,111\}$ are constructed. The search uses specific partitioning of columns of the orbit incidence matrix, related to so-called ``quasi-designs''. Actions of groups $PSL(2,25)$, $PGL(2,25)$ and twisted $PGL(2,25)$ are being compared. It is shown that there exist $4$-$(26,6,\lambda)$ designs with $PGL(2,25)$, respectively twisted $PGL(2,25)$ as a group of automorphisms and with $\lambda$ in the set $\{51,60,81,90,111\}$. With $\lambda$ in the set $\{60,81\}$, there exist designs which possess all three considered groups as groups of automorphisms. An overview of $t$-$(q+1,k,\lambda)$ designs with $PSL(2,q)$ as group of automorphisms and with $(t,k) \in \{(4,5), (4,6), (5,6)\}$ is included.
References:
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